Method for computing the intensity of specularly reflected light

ABSTRACT

The intensity of specularly reflected light from an illuminated object is represented by an algebraic expression including multiplication, addition, and subtraction operations. The algebraic expression is used in an illumination model, where the illumination model describes the color and intensity of light reflected by the illuminated object. Light reflected by the illuminated object is composed of ambient, diffuse, and specular components. The specular terms used in the illumination model are equivalent in functional form to the diffuse terms, thereby accelerating the computation of color vector c defined by the illumination model. A modified algebraic expression representing specularly reflected light from an illuminated object is defined and used in the illumination model, thereby accelerating computation of color vector c.

CROSS REFERENCE TO RELATED APPLICATION

This application is a continuation application and claims the prioritybenefit of U.S. patent application Ser. No. 09/935,123 entitled “Methodfor Computing the Intensity of Specularly Reflected Light” filed Aug.21, 2001 and now U.S. Pat. No. 6,781,594. The disclosure of thiscommonly owned and assigned application is incorporated herein byreference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates generally to computer generated images and moreparticularly to a method for computing the intensity of specularlyreflected light.

2. Description of the Background Art

The illumination of a computer-generated object by colored light sourcesand ambient light is described by an illumination model. Theillumination model is a mathematical expression including ambient,diffuse, and specular illumination terms. The object is illuminated bythe reflection of ambient light and the reflection of light source lightfrom the surface of the object. Therefore, the illumination of theobject is composed of ambient, diffuse, and specularly reflected light.Given ambient light and light sources positioned about the object, theillumination model defines the reflection properties of the object. Theillumination model is considered to be accurate if the illuminatedobject appears realistic to an observer. Typically, the illuminationmodel is incorporated in a program executed by a rendering engine, avector processing unit, or a central processing unit (CPU). The programmust be capable of computing the illumination of the object when thelight sources change position with respect to the object, or when theobserver views the illuminated object from a different angle, or whenthe object is rotated. Furthermore, an efficient illumination model isneeded for the program to compute the illumination in real-time, forexample, if the object is rotating. Therefore, it is desired toincorporate terms in the illumination model that are computationallycost effective, while at the same time generating an image of theilluminated object that is aesthetically pleasing to the observer.

Ambient light is generalized lighting not attributable to direct lightrays from a specific light source. In the physical world, for example,ambient light is generated in a room by multiple reflections of overheadflorescent light by the walls and objects in the room, providing anomni-directional distribution of light. The illumination of the objectby ambient light is a function of the color of the ambient light and thereflection properties of the object.

The illumination of the object by diffuse and specular light dependsupon the colors of the light sources, positions of the light sources,the reflection properties of the object, the orientation of the object,and the position of the observer. Source light is reflected diffuselyfrom a point on the object's surface when the surface is rough,scattering light in all directions. Typically, the surface is consideredrough when the scale length of the surface roughness is approximatelythe same or greater than the wavelength of the source light. FIG. 1Aillustrates diffuse reflection from an object's surface. A light ray i105 from a source 110 is incident upon a surface 115 at point P 120,where a bold character denotes a vector. Light ray i 105 is scattereddiffusely about point P 120 into a plurality of light rays r₁ 125, r₂125, r₃ 125, r₄ 125, and r₅ 125.

If the scale length of the surface roughness is much less than thewavelength of the source light, then the surface is considered smooth,and light is specularly reflected. Specularly reflected light is notscattered omni-directionally about a point on the object's surface, butinstead is reflected in a preferred direction. FIG. 1B illustratesspecular reflection from an object's surface. A light ray i 130 from asource 135 is incident upon a surface 140 at a point P 145. Light ray i130 is specularly reflected about point P 145 into a plurality of lightrays r₁ 150, r₂ 150, r₃ 150, r₄ 150, and r₅ 155, confined within a cone160 subtended by angle φ 165. Light ray r 155 is the preferred directionfor specular reflection. That is, the intensity of specularly reflectedlight has a maximum along light ray r 155. As discussed further below inconjunction with FIGS. 2A–2B, the direction of preferred light ray r 155is specified when the angle of reflection is equal to the angle ofincidence.

Typically, objects reflect light diffusely and specularly, and in orderto generate a realistic illumination of the computer-generated objectthat closely resembles the real physical object, both diffuse andspecular reflections need to be considered.

FIG. 2A illustrates specular reflection from an object's surface in apreferred direction, including a unit vector l 205 pointing towards alight source 210, a unit vector n 215 normal to a surface 220 at a pointof reflection P 225, a unit vector r 230 pointing in the preferredreflected light direction, a unit vector v 235 pointing towards anobserver 240, an angle of incidence θ_(i) 245 subtended by the unitvector l 205 and the unit vector n 215, an angle of reflection θ_(r) 250subtended by unit vector n 215 and the unit vector r 230, and an angleθ_(rv) 255 subtended by unit vector r 230 and unit vector v 235. Lightfrom the source 210 propagates in the direction of a unit vector −l 260,and is specularly reflected from the surface 220 at point P 225. A unitvector is a vector of unit magnitude.

Reflection of light from a perfectly smooth surface obeys Snell's law.Snell's law states that the angle of incidence θ_(i) 245 is equal to theangle of reflection θ_(r) 250. If surface 220 is a perfectly smoothsurface, light from source 210 directed along the unit vector −l 260 atan angle of incidence θ_(i) 245 is reflected at point P 225 along unitvector r 230 at an angle of reflection θ_(r) 250, where θ_(i)=θ_(r).Consequently, if surface 220 is a perfectly smooth surface, then lightdirected along −l 260 from source 210 and specularly reflected at pointP 225 would not be detected by observer 240, since specularly reflectedlight is directed only along unit vector r 230. However, a surface isnever perfectly smooth, and light directed along −l 260 from source 210and specularly reflected at point P 225 has a distribution about unitvector r 230, where unit vector r 230 points in the preferred directionof specularly reflected light. The preferred direction is specified byequating the angle of incidence θ_(i) 245 with the angle of reflectionθ_(r) 250. In other words, specular reflection intensity as measured byobserver 240 is a function of angle θ_(rv) 255, having a maximumreflection intensity when θ_(rv)=0 and decreasing as θ_(rv) 255increases. That is, observer 240 viewing point P 225 of the surface 220detects a maximum in the specular reflection intensity when unit vectorv 235 is co-linear with unit vector r 230, but as observer 240 changesposition and angle θ_(rv) 235 increases, observer 240 detects decreasingspecular reflection intensities.

A first prior art method for computing the intensity of specularlyreflected light is to represent the specular intensity asf(r,v,n)∝(r·v)^(n), where 1≦n≦∞ and n is a parameter that describes theshininess of the object. Since r and v are unit vectors, the dot productr·v=cos θ_(rv), and therefore, f(r,v,n)∝cos^(nθ) _(rv).

A second prior art method computes the intensity of specularly reflectedlight in an alternate manner. For example, FIG. 2B illustrates anotherembodiment of specular reflection from an object's surface in apreferred direction, including a unit vector l 265 pointing towards alight source 270, a unit vector n 275 normal to a surface 280 at a pointof reflection P 282, a unit vector r 284 pointing in the preferredreflected light direction, a unit vector v 286 pointing towards anobserver 288, a unit vector h 290 bisecting the angle subtended by theunit vector l 265 and the unit vector v 286, an angle of incidence θ_(i)294, an angle of reflection θ_(r) 290, and an angle θ_(nh) 292 subtendedby the unit vector h 290 and the unit vector n 275. Light from thesource 270 propagates in the direction of a unit vector −l 272. Theangle of incidence θ_(i) 294 is equal to the angle of reflection θ_(r)290. The specular intensity is represented as g(n,h,n)∝(n·h)^(n), where1≦n≦∞ and n is a parameter that describes the shininess of the object.Since n and h are unit vectors, the dot product n·h=cos θ_(nh), andtherefore, g(n,h,n)∝cos^(n) θ_(nh). When the surface 280 is rotated suchthat unit vector n 275 is co-linear with unit vector h 290, then cosθ_(nh)=1, the specular intensity g(n,h,n) is at a maximum, and thereforethe observer 288 detects a maximum in the specularly reflected lightintensity. The second prior art method for computing the intensity ofspecular reflection has an advantage over the first prior art method inthat the second prior art method more closely agrees with empiricalspecular reflection data.

The first and second prior art methods for computing the intensity ofspecularly reflected light are computationally expensive compared to thecalculation of the diffuse and ambient terms that make up the remainderof the illumination model. Specular intensity as defined by the priorart is proportional to cos^(n) θ, where θ≡θ_(rv) or θ≡θ_(nh). Theexponential specular intensity function cos^(n) θ can be evaluated forinteger n, using n−1 repeated multiplications, but this is impracticalsince a typical value of n can easily exceed 100. If the exponent n isequal to a power of two, for example n=2^(m), then the specularintensity may be calculated by m successive squarings. However, theevaluation of specular intensity is still cost prohibitive. If n is notan integer, then the exponential and logarithm functions can be used, byevaluating cos^(n) θ as e^((n ln(cos θ))), but exponentiation is atleast an order of magnitude slower than the operations required tocompute the ambient and diffuse illumination terms.

A third prior art method of computing specular intensity is to replacethe exponential specular intensity function with an alternate formulathat invokes a similar visual impression of an illuminated object,however without exponentiation. Specular intensity is modeled by analgebraic function h(t,n)=t /(n−nt+t), where either t=cos θ_(rv) ort=cos θ_(nh), and n is a parameter that describes the shininess of theobject. The algebraic function h(t,n) does not include exponents, butdoes include multiplication, addition, subtraction, and divisionoperators. These algebraic operations are usually less costly thanexponentiation. However, while the computation time has been reduced, inmany computer architectures division is still the slowest of theseoperations.

It would be useful to implement a cost effective method of calculatingspecular intensity that puts the computation of the specular term on amore even footing with the computation of the ambient and diffuse terms,while providing a model of specular reflection that is aestheticallypleasing to the observer.

SUMMARY OF THE INVENTION

In accordance with the present invention, an algebraic method isdisclosed to compute the intensity of specularly reflected light from anobject illuminated by a plurality of light sources. The plurality oflight sources include point light sources and extended light sources.The algebraic expression S_(i)(n,h_(i),n)=1−n+max{n·(nh_(i)), n−1}represents the intensity of light reflected from a point on the objectas measured by an observer, the object illuminated by an i^(th) lightsource. The algebraic expression includes multiplication, addition, andsubtraction operators. The algebraic expression approximates the resultsof prior art models of specular reflection intensity, but at lowercomputational costs.

The algebraic expression for specular intensity is substituted into anillumination model, where the illumination model includes ambient,diffuse, and specular illumination terms. The illumination model isincorporated into a software program, where the program computes a colorvector c representing the color and intensity of light reflected by anobject illuminated by a plurality of light sources. The reflected lightis composed of ambient, diffuse, and specular components. The specularterms in the illumination model are equivalent in functional form to thediffuse terms, thereby providing an efficient and inexpensive means ofcomputing the specular component of the color vector c. That is, avector-based hardware system that computes and sums the ambient anddiffuse terms can be used to compute and sum the ambient, diffuse, andspecular terms at very little additional cost.

A modified algebraic expression SM_(i,k)(n,h_(i),n)=(1−n/k+max{n·(n/kh_(i)), n/k−1})^(k) represents the intensity of light reflected from apoint on the object, where the object is illuminated by the i^(th) lightsource, and 2≦k≦n. The first (k−1) derivatives of the modified algebraicexpression SM_(i,k)(n,h_(i),n) are continuous, and therefore byincreasing the value of k, the modified algebraic expression moreclosely approximates the prior art specular intensity functions, but ata lower computational cost.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A of the prior art illustrates diffuse reflection from an object'ssurface;

FIG. 1B of the prior art illustrates specular reflection from anobject's surface;

FIG. 2A of the prior art illustrates specular reflection from anobject's surface in a preferred direction;

FIG. 2B of the prior art illustrates another embodiment of specularreflection from an object's surface in a preferred direction;

FIG. 3 is a block diagram of one embodiment of an electronicentertainment system in accordance with the invention;

FIG. 4A is a graph of the specular intensity function S(n,h,n) accordingto the invention, the specular intensity function g(n,h,n) of the priorart, and the specular intensity function h(n,h,n) of the prior art, forn=3;

FIG. 4B is a graph of the specular intensity function S(n,h,n) accordingto the invention, the specular intensity function g(n,h,n) of the priorart, and the specular intensity function h(n,h,n) of the prior art, forn=10;

FIG. 4C is a graph of the specular intensity function S(n,h,n) accordingto the invention, the specular intensity function g(n,h,n) of the priorart, and the specular intensity function h(n,h,n) of the prior art, forn=50;

FIG. 4D is a graph of the specular intensity function S(n,h,n) accordingto the invention, the specular intensity function g(n,h,n) of the priorart, and the specular intensity function h(n,h,n) of the prior art, forn=200;

FIG. 5 illustrates preferred directions of specular reflection for twoorientations of a surface, according to the invention;

FIG. 6 illustrates illumination of an object by a plurality of lightsources, according to the invention;

FIG. 7 illustrates one embodiment of color vector c in (R,G,B)-space,according to the invention;

FIG. 8A is a graph of the prior art specular intensity functiong(n,h,n), the specular intensity function S(n,h,n) according to theinvention, and the modified specular intensity function SM₂(n,h,n)according to the invention, for n=3;

FIG. 8B is a graph of the prior art specular intensity functiong(n,h,n), the specular intensity function S(n,h,n) according to theinvention, the modified specular intensity function SM₂(n,h,n) accordingto the invention, the modified specular intensity function SM₄(n,h,n)according to the invention, and the modified specular intensity functionSM₈(n,h,n) according to the invention, for n=10;

FIG. 8C is a graph of the prior art specular intensity functiong(n,h,n), the specular intensity function S(n,h,n) according to theinvention, the modified specular intensity function SM₂(n,h,n) accordingto the invention, the modified specular intensity function SM₄(n,h,n)according to the invention, and the modified specular intensity functionSM₈(n,h,n) according to the invention, for n=50; and

FIG. 8D is a graph of the prior art specular intensity functiong(n,h,n), the specular intensity function S(n,h,n) according to theinvention, the modified specular intensity function SM₂(n,h,n) accordingto the invention, the modified specular intensity function SM₄(n,h,n)according to the invention, and the modified specular intensity functionSM₈(n,h,n) according to the invention, for n=200;

DETAILED DESCRIPTION OF THE DRAWINGS

FIG. 3 is a block diagram of one embodiment of an electronicentertainment system 300 in accordance with the invention. System 300includes, but is not limited to, a main memory 310, a central processingunit (CPU) 312, a vector processing unit VPU 313, a graphics processingunit (GPU) 314, an input/output processor (IOP) 316, an IOP memory 318,a controller interface 320, a memory card 322, a Universal Serial Bus(USB) interface 324, and an IEEE 1394 interface 326. System 300 alsoincludes an operating system read-only memory (OS ROM) 328, a soundprocessing unit (SPU) 332, an optical disc control unit 334, and a harddisc drive (HDD) 336, which are connected via a bus 346 to IOP 316.

CPU 312, VPU 313, GPU 314, and IOP 316 communicate via a system bus 344.CPU 312 communicates with main memory 310 via a dedicated bus 342. VPU313 and GPU 314 may also communicate via a dedicated bus 340.

CPU 312 executes programs stored in OS ROM 328 and main memory 310. Mainmemory 310 may contain prestored programs and may also contain programstransferred via IOP 316 from a CD-ROM or DVD-ROM (not shown) usingoptical disc control unit 334. IOP 316 controls data exchanges betweenCPU 312, VPU 313, GPU 314 and other devices of system 300, such ascontroller interface 320.

Main memory 310 includes, but is not limited to, a program having gameinstructions including an illumination model. The program is preferablyloaded from a CD-ROM via optical disc control unit 334 into main memory310. CPU 312, in conjunction with VPU 313, GPU 314, and SPU 332,executes game instructions and generates rendering instructions usinginputs received via controller interface 320 from a user. The user mayalso instruct CPU 312 to store certain game information on memory card322. Other devices may be connected to system 300 via USB interface 324and IEEE 1394 interface 326.

VPU 313 executes instructions from CPU 312 to generate color vectorsassociated with an illuminated object by using the illumination model.SPU 332 executes instructions from CPU 312 to produce sound signals thatare output on an audio device (not shown). GPU 314 executes renderinginstructions from CPU 312 and VPU 313 to produce images for display on adisplay device (not shown). That is, GPU 314, using the color vectorsgenerated by VPU 313 and rendering instructions from CPU 312, rendersthe illuminated object in an image.

The illumination model includes ambient, diffuse, and specularillumination terms. The specular terms are defined by substituting aspecular intensity function into the illumination model. In the presentinvention, specular intensity is modeled by the function S, whereS(n,h,n)=1−n+max{n·(nh), n−1} and the function max{n·(nh), n−1} selectsthe maximum of n·(nh) and n−1. The unit vector n 275 and the unit vectorh 290 are described in conjunction with FIG. 2B, and n is the shininessparameter. When unit vector n 275 is co-linear with unit vector h 290and θ_(nh)=0, max{n·(nh), n−1}=max{n, n−1}=n, and S(n,h,n)=(1−n)+n=1. Inother words, S is at a maximum when unit vector n 275 is co-linear withunit vector h 290. However, when unit vector n 275 is not co-linear withunit vector h 290 and when the condition n·(nh)≦n−1 is satisfied, thenmax{n·(nh), n−1}=n−1, and S(n,h,n)=(1−n)+(n−1)=0. In other words,S(n,h,n)=0 when n·(nh)≦n−1. Since n·(nh)=n(cos θ_(nh)), S(n,h,n)=0 whencos θ_(nh)≦1−1/n. That is, S(n,h,n)=0 for θ_(nh)≧|arcos(1−1/n)| and forθ_(nh)≦−|arcos(1−1/n)|, where the function |(arg)| generates theabsolute value of the argument (arg). In contrast to the prior artspecular intensity functions, specular intensity function S does notinclude exponentiation nor does function S include a division.Therefore, computing function S is less costly than computing the priorart specular intensity functions. In addition, the graph of function Sis similar to the prior art specular intensity functions, therebyproviding a reasonable model for specular reflection. For example, FIG.4A is a graph of the specular intensity function S(n,h,n) 410 accordingto the present invention, the second prior art specular intensityfunction g(n,h,n) 420, and the third prior art specular intensityfunction h(n,h,n) 430, plotted as functions of angle θ_(nh) 292. Theshininess parameter n=3. All curves have maximum specular intensity whenθ_(nh)=0. The maximum specular intensity of each curve is equal to 1.0,and the minimum specular intensity of each curve is equal to zero. Inaddition, each curve is a continuous function of θ_(nh). FIG. 4B is agraph of the functions S(n,h,n) 410, g(n,h,n) 420, and h(n,h,n) 430 forn=10, FIG. 4C is a graph of the functions for n=50, and FIG. 4D is agraph of the functions for n=200. Note that as the shininess parameter nincreases, the width of each function decreases, where the width of eachfunction can be measured at a specular intensity value of 0.5, forexample. The relation between the width of the specular intensityfunction and the shininess parameter n is explained further below inconjunction with FIG. 5.

FIG. 5 illustrates preferred directions of specular reflection for twoorientations of a surface. Unit vector l 505 points towards a lightsource 501. Light travels from the light source 501 along a unit vector−l 502, and is reflected from point P 510. Unit vector v 515 pointstowards an observer (not shown). Unit vector v 515 and unit vector l 505are constant since source 401 and the observer (not shown) arestationary. Given a first orientation of an object 520 with a unitvector n₁ 525 normal to a surface 530, light directed along −l 502 ismaximally specularly reflected along a unit vector r₁ 535. Given asecond orientation of the object 540 with a unit vector n₂ 545 normal toa surface 550, light directed along −l 502 is maximally specularlyreflected along a unit vector r₂ 555. Angle φ560 defines the regionabout unit vector v 515 in which the intensity of specularly reflectedlight measured by the observer located along unit vector v 515 isgreater than 0.5. If object 520 is rotated such that unit vector r₁ 535is directed along unit vector v 515, then the observer measures amaximum specular intensity of 1.0. If n is large, then the material isextremely shiny, φ560 is small, and the specularly reflected light isconfined to a relatively narrow region about unit vector v 515. If n issmall, then the material is less shiny, φ560 is large, and thespecularly reflected light is confined to a wider region about unitvector v 515.

Therefore, specular intensity function S according to the presentinvention properly models the shininess of an object as embodied in theshininess parameter n.

As will be discussed further below in conjunction with FIG. 7, anadditional advantage of the present invention's model for specularlyreflected light intensity is the similarity in form of the specularterms to the ambient and diffuse terms in the illumination model. Thus,a vector-based hardware system that computes and sums the ambient anddiffuse terms can be used to compute and sum the ambient, diffuse, andspecular terms at very little additional cost.

FIG. 6 illustrates illumination of an object by a plurality of lightsources, including a unit vector l_(i) 605 pointing towards an i^(th)light source 610, a unit vector n 615 normal to a surface 620 at a pointof reflection P 625, a unit vector v 630 pointing towards an observer635, a unit vector r_(i) 640 pointing in the preferred specularreflection direction, and a unit vector h_(i) 645 bisecting the angleφ_(i) 650 subtended by the unit vector l_(i) 605 and the unit vector v630. In addition, FIG. 6 includes a plurality of point light sources 655and an extended light source 660. Only one extended light source 660 isshown, although the scope of the invention encompasses a plurality ofextended light sources. The extended light source 660 is composed of aplurality of point light sources 665. Specular intensity is modeled by afunction S_(i)=1−n+max{n·(nh_(i)), n−1}, according to the presentinvention, where S_(i) is the specular intensity of point P 625illuminated by the i^(th) light source 610, and detected by the observer635.

Diffuse intensity is modeled by the function D_(i)=max{n·l_(i)0} of theprior art, where D_(i) is the diffuse intensity of point P 625illuminated by the i^(th) light source 610, and detected by observer635. Given specular intensity S_(i), diffuse intensity D_(i), andambient light, an illumination model for a resulting color vector c isdefined. The color vector c is a vector in (R,G,B)-space and is used todescribe the resulting color and light intensity of an illuminatedobject viewed by an observer, due to specular, diffuse, and ambientlight reflection.

FIG. 7 illustrates one embodiment of color vector c 705 in(R,G,B)-space, defined by a Cartesian coordinate system including a red(R) axis 710, a green (G) axis 715, and a blue (B) axis 720. Colorvector c 705 has a component c_(R) 725 along the R axis 710, a componentc_(G) 730 along the G axis 715, and a component c_(B) 735 along the Baxis 720. The value of the components c_(R) 725, c_(G) 730, and c_(B)735 of color vector c 705 determine the color of the reflected light,and the magnitude of color vector c 705 determines the intensity of thereflected light.

Given M light sources, the equation for color vector c 705 of thepresent invention is written as

${c = {{k_{d} \otimes \left( {c_{a} + {\sum\limits_{i = 1}^{M}\;{D_{i}c_{i}}}} \right)} + {k_{s} \otimes {\sum\limits_{i = 1}^{M}\;{S_{i}c_{i}}}}}},$where k_(d), k_(s) are diffuse and specular reflection coefficientvectors, respectively, c_(a) is the color of the ambient light, andc_(i) is the color of the i^(th) light source 610. Vectors c_(a) andc_(i) are vectors in (R,G,B)-space. The summation symbol is used to sumthe index i over all M light sources, and the symbol {circle around (×)}operates on two vectors to give the component-wise product of the twovectors.

Substituting the diffuse intensity D_(i) and the specular intensityS_(i) into the equation for color vector c, and rearranging terms, theequation can be written as

$c = {a + {\sum\limits_{j = 1}^{2}\;{\sum\limits_{i = 1}^{M}\;{{\max\left( {{n \cdot u_{ji}},m_{ji}} \right)}\;{c_{ji}.}}}}}$The vector a includes an ambient color vector. The sum over index i forj=1 is a summation over all the diffuse color vectors generated by the Mlight sources, and the sum over index i for j=2 is a summation over allthe specular color vectors generated by the M light sources. Forexample, u_(1i)=l_(i), m_(1i)=0, and c_(1i)=k_(d){circle around(×)}c_(i) are the diffuse terms that generate the diffuse color vectorfor the i^(th) light source, and u_(2i)=nh_(i), m_(2i)=n−1, andc_(2i)=k_(s){circle around (×)}c_(i) are the specular terms thatgenerate the specular color vector for the i^(th) light source.

Each term in the summation over i and j in the expression for the colorvector c has the same form. In other words, the specular illuminationterm max(n·u_(2i), m_(2i)) for the i^(th) light source has the samefunctional form as the diffuse illumination term max(n·u_(1i), m_(1i)),and therefore, using vector-based computer hardware, such as VPU 313 ofFIG. 3, the vector dot products for the diffuse and specular terms canbe evaluated in parallel, providing an efficient means of computing thespecular terms. Compared to the computation of the diffuse terms alone,very little overhead is needed to compute the specular and diffuse termstogether. In addition, since summation is not a very costly operation,little overhead is needed to sum the specular and diffuse terms incomputing the color vector c, since instead of summing M diffuse terms,a sum is made over M diffuse terms and M specular terms.

In addition, if for each light source i 610, the light source directionvector l_(i) 605, the observer vector v 630, the shininess index n, thecolor of the light c_(i), and the diffuse and specular reflectioncoefficients k_(d), k_(s), respectively, are constant over surface 620of object 670, which is a valid assumption for certain circumstancessuch as when light source i 610 and observer 635 are placed far fromobject 670, then variables u_(1i), m_(1i), c_(1i), u_(2i), m_(2i), andc_(2i) only need to be calculated once for each light source i 610.Consequently, the additional cost introduced by the calculation of thespecular and diffuse terms in comparison to the calculation of only thediffuse terms is very negligible, since the calculation of the specularand diffuse reflected light from every point on surface 620 of object670 only involves a parallel computation of the vector dot products anda sumation of 2M diffuse and specular terms given M light sources. Thecalculation does not involve a computation of variables u_(1i), m_(1i),c_(1i), u_(2i), m_(2i), and c_(1i) at every point on surface 620 ofobject 670 for each light source i 610. In other words, for a givenlight source i 610, the unit vector n 615 is the only component of thecolor equation that is variable over surface 620 of object 670, andhence the calculation of the specular terms are “almost free” incomparison to the calculation of the diffuse terms alone.

As illustrated in FIGS. 4A–4D, specular intensity function S(n,h,n) 410of the present invention is not continuous in the first derivative. Thatis, there is a discontinuity in the first derivative of S with respectto θ_(nh) when S=0. Since S=1−n+max{n·nh, n−1), the discontinuity occurswhen n·nh=n−1, or when cos θ_(nh)=1−1/n. For example, referring to FIG.4A, the discontinuity in the first derivative of S with respect toθ_(nh) for n=3 occurs when cos θ_(nh)=1−1/3, or in other words, whenθ_(nh)=∓48.2 degrees. In order to more closely approximate the prior artspecular intensity function g(n,h,n) 420, a modified specular intensityfunction SM₂(n,h,n)≡S²(n,h,n/2) is defined according to the presentinvention. Modified specular intensity function SM₂(n,h,n) has acontinuous first derivative. However, the modified specular intensityfunction SM₂(n,h,n) has discontinuous higher order derivatives. Forexample, the second order derivative is discontinuous. Another modifiedspecular intensity function according to the present invention,SM₃(n,h,n)≡S³(n,h,n/3), has continuous first and second orderderivatives, but has discontinuous higher order derivatives. In fact,the modified specular intensity functionSM_(k)(n,h,n)≡S^(k)(n,h,n/k)=(1−n/k+max{n·(n/k h), n/k−1})^(k) of thepresent invention, where 2≦k≦n, has continuous derivatives up to andincluding the (k−1)th order derivative. For k=n, the modified specularintensity function SM_(n)(n,h,n)=max{n·h, 0}^(n)=cos^(n) θ_(nh), and isequivalent to the prior art specular intensity function g(n,h,n) 420.Therefore, the modified specular intensity function SM_(k)(n,h,n)according to the present invention, where 2≦k≦n, can more closelyapproximate the prior art specular intensity function g(n,h,n) 420 byincreasing the value of k.

Referring to FIG. 6, the modified specular intensity function is definedfor an object illuminated by a plurality of light sources, whereSM_(i,k)(n,h_(i),n)=(1−n/k+max{n·(n/k h_(i)), n/k−1})^(k) is themodified specular intensity function for the object 670 illuminated bythe i^(th) light source 610.

FIG. 8A is a graph of the prior art specular intensity function g(n,h,n)420, the specular intensity function S(n,h,n) 410 of the presentinvention, and the modified specular intensity function SM₂(n,h,n) 800of the present invention, for parameter n=3. Similarly, each of FIGS.8B-8D is a graph of the prior art specular intensity function g(n,h,n)420, the specular intensity function S(n,h,n) 410 of the presentinvention, the modified specular intensity function SM₂(n,h,n) 800 ofthe present invention, the modified specular intensity functionSM₄(n,h,n) 810 of the present invention, and the modified specularintensity function SM₈(n,h,n) 820 of the present invention, where n=10in FIG. 8B, n=50 in FIG. 8C, and n=200 in FIG. 8D. Each of FIGS. 8A–8Dillustrate that as the parameter k of the modified specular intensityfunction SM_(k)(n,h,n) increases, the graph of function SM_(k)(n,h,n)approaches the graph of the prior art specular intensity functiong(n,h,n) 420. In addition, the modified function SM₂(n,h,n) 800 has acontinuous first-order derivative, the modified function SM₄(n,h,n) 810has continuous derivatives up to third-order, and the modified functionSM₈(n,h,n) 820 has continuous derivatives up to seventh-order.Therefore, the modified specular intensity function SM_(k)(n,h,n) can beused to more closely approximate the prior art specular intensityfunction g(n,h,n) 420, however, at a lower cost than computing the priorart specular intensity function g(n,h,n) 420. That is, SM_(k)(n,h,n) canbe evaluated inexpensively when k is a small power of 2, by successivemultiplications. As FIG. 8D illustrates for n=200, a good approximationto the prior art specular intensity function g(n,h,n)∝cos²⁰⁰ θ_(nh) isachieved by using the modified specular intensity function SM₈(n,h,n)820 according to the present invention. SM₈(n,h,n) 820 can be computedwith three successive multiplications. For example,SM₈(n,h,n)=f₂(n,h,n/8)×f₂(n,h,n/8), wheref₂(n,h,n/8)=f₁(n,h,n/8)×f₁(n,h,n/8), and wheref₁(n,h,n/8)=S(n,h,n/8)×S(n,h,n/8). In contrast, the computation of theprior art specular intensity function g(n,h,n)∝cos²⁰⁰ θ_(nh) requiresexponentiation to the 200_(th) power, which is a more costlycomputation.

The invention has been explained above with reference to preferredembodiments. Other embodiments will be apparent to those skilled in theart in light of this disclosure. The present invention may readily beimplemented using configurations other than those described in thepreferred embodiments above. For example, the program including theillumination model, according to the invention, may be executed in partor in whole by the CPU, the VPU, the GPU, or a rendering engine (notshown). Additionally, the present invention may effectively be used inconjunction with systems other than the one described above as thepreferred embodiment. Therefore, these and other variations upon thepreferred embodiments are intended to be covered by the presentinvention, which is limited only by the appended claims.

1. A method for computing the intensity of specularly reflected light,comprising the steps of: representing the intensity of light reflectedspecularly from an object illuminated by a plurality of light sources byan algebraic expression, wherein the algebraic expression is S_(i)(n,h_(i),n)=1−n+max{n·(nh_(i)), n−1}, which describes the intensity oflight reflected from a point on the object as measured by an observer,the object illuminated by light from an i^(th) light source, where n isa unit vector normal to the object at the point of reflection, h_(i) isa unit vector bisecting an angle subtended by a unit vector pointingtowards the i^(th) light source from the point of reflection and a unitvector pointing towards the observer from the point of reflection, and nis a parameter that describes the shininess of the object; incorporatingthe algebraic expression into an illumination model for the illuminationof the object, the model having at least specular illumination terms anddiffuse illumination terms; and expressing the specular illuminationterms of the illumination model in the same functional form as otherterms of the illumination model, wherein the specular illumination termsand the diffuse illumination terms are evaluated substantially inparallel.
 2. The method of claim 1, wherein the algebraic expressiondoes not include division or exponentiation operators.
 3. The method ofclaim 1, wherein the plurality of light sources includes extended lightsources and point light sources.
 4. The method of claim 1, wherein theillumination model describes the color and intensity of light reflectedfrom the object illuminated by the i^(th) light source, the reflectedlight including specular, diffuse, and ambient components.
 5. The methodof claim 4, wherein the other terms of the illumination model furthercomprise ambient terms.
 6. The method of claim 1, wherein the specularillumination terms of the illumination model are expressed in the samefunctional form as the diffuse illumination terms of the illuminationmodel.
 7. A method for computing the intensity of specularly reflectedlight, comprising the steps of: representing the intensity of lightreflected specularly from an object illuminated by a plurality of lightsources by an algebraic expression, wherein the algebraic expression isSM_(i,k)(n,h_(i),n)=(1−n/k+max{n·(n/k h_(i)), n/k−1})^(k), whichdescribes the intensity of light reflected from a point on the object asmeasured by an observer, the object illuminated by light from an i^(th)light source, where n is a unit vector normal to the object at the pointof reflection, h_(i) is a unit vector bisecting an angle subtended by aunit vector pointing towards the i^(th) light source from the point ofreflection and a unit vector pointing towards the observer from thepoint of reflection, n is a parameter that describes the shininess ofthe object, and k is a parameter that determines which derivatives ofthe algebraic expression are continuous; incorporating the algebraicexpression into an illumination model for the illumination of theobject, the model having at least specular illumination terms anddiffuse illumination terms; and expressing the specular illuminationterms of the illumination model in the same functional form as otherterms of the illumination model, wherein the specular illumination termsand the diffuse illumination terms are evaluated substantially inparallel.
 8. The method of claim 7, wherein 2≦k≦n.
 9. Acomputer-readable medium comprising instructions for computing theintensity of specularly reflected light by performing the steps of:representing the intensity of light reflected specularly from an objectilluminated by a plurality of light sources by an algebraic expression,wherein the algebraic expression is S_(i)(n,h_(i),n)=1−n+max{n·(nh_(i)),n−1}, which describes the intensity of light reflected from a point onthe object as measured by an observer, the object illuminated by lightfrom an i^(th) light source, where n is a unit vector normal to theobject at the point of reflection, h_(i) is a unit vector bisecting anangle subtended by a unit vector pointing towards the i^(th) lightsource from the point of reflection and a unit vector pointing towardsthe observer from the point of reflection, and n is a parameter thatdescribes the shininess of the object; incorporating the algebraicexpression into an illumination model for the illumination of theobject, the model having at least specular illumination terms anddiffuse illumination terms; and expressing the specular illuminationterms of the illumination model in the same functional form as otherterms of the illumination model, wherein the specular illumination termsand a diffuse terms are evaluated substantially in parallel.
 10. Thecomputer-readable medium of claim 9, wherein the algebraic expressiondoes not include division or exponentiation operators.
 11. Thecomputer-readable medium of claim 9, wherein the plurality of lightsources includes extended light sources and point light sources.
 12. Thecomputer-readable medium of claim 9, wherein the illumination modeldescribes the color and intensity of light reflected from the objectilluminated by the i^(th) light source, the reflected light includingspecular, diffuse, and ambient components.
 13. The computer-readablemedium of claim 12, wherein the other terms of the illumination modelfurther comprise ambient terms.
 14. The computer-readable medium ofclaim 9, wherein the specular illumination terms of the illuminationmodel are expressed in the same functional form as the diffuseillumination terms of the illumination model.
 15. A computer-readablemedium comprising instructions for computing the intensity of specularlyreflected light by performing the steps of: representing the intensityof light reflected specularly from an object illuminated by a pluralityof light sources by an algebraic expression, wherein the algebraicexpression is SM_(i,k)(n,h_(i),n)=(1−n/k+max{n·(n/k h_(i)), n/k−1})^(k), which describes the intensity of light reflected from a point on theobject as measured by an observer, the object illuminated by light froman i^(th) light source, where n is a unit vector normal to the object atthe point of reflection, h_(i) is a unit vector bisecting an anglesubtended by a unit vector pointing towards the i^(th) light source fromthe point of reflection and a unit vector pointing towards the observerfrom the point of reflection, n is a parameter that describes theshininess of the object, and k is a parameter that determines whichderivatives of the algebraic expression are continuous; incorporatingthe algebraic expression into an illumination model for the illuminationof the object, the model having at least specular illumination terms anddiffuse illumination terms; and expressing the specular illuminationterms of the illumination model in the same functional form as otherterms of the illumination model, wherein the specular illumination termsand a diffuse terms are evaluated substantially in parallel.
 16. Thecomputer-readable medium of claim 15, wherein 2≦k≦n.
 17. An electronicentertainment system for computing the illumination of an object by aplurality of light sources, comprising: a memory configured to storegame instructions and an illumination model; a processor configured toexecute game instructions and generate rendering instructions; a vectorprocessor configured to calculate color vectors using the illuminationmodel, the illumination model having specular illumination terms anddiffuse illumination terms expressed in the same functional form,wherein the vector processor evaluates vector dot products for thediffuse and specular illumination terms substantially in parallel; and agraphics processor configured to render the illuminated object in animage using the color vectors according to the rendering instructions,wherein for each light source i, an algebraic expression representingthe intensity of light reflected specularly from a point on the objectand detected by an observer is substituted into the illumination modelyielding a specular illumination term for the light source i, whereinthe algebraic expression for light source i isS_(i)(n,h_(i),n)=1−n+max{n·(nh_(i)), n−1}, where n is a unit vectornormal to the object at the point of reflection, h_(i) is a unit vectorbisecting an angle subtended by a unit vector pointing towards lightsource i from the point of reflection and a unit vector pointing towardsthe observer from the point of reflection and n is a parameter thatdescribes the shininess of the object.
 18. The electronic entertainmentsystem of claim 17, wherein the algebraic expression does not containdivision or exponentiation operators.
 19. An electronic entertainmentsystem for computing the illumination of an object by a plurality oflight sources, comprising: a memory configured to store gameinstructions and an illumination model; a processor configured toexecute game instructions and generate rendering instructions; a vectorprocessor configured to calculate color vectors using the illuminationmodel, the illumination model having specular illumination terms anddiffuse illumination terms expressed in the same functional form,wherein the vector processor evaluates vector dot products for thediffuse and specular illumination terms substantially in parallel; and agraphics processor configured to render the illuminated object in animage using the color vectors according to the rendering instructions,wherein for each light source i, an algebraic expression representingthe intensity of light reflected specularly from a point on the objectand detected by an observer is substituted into the illumination modelyielding a specular illumination term for the light source i, whereinthe algebraic expression for light source i isSM_(i,k)(n,h_(i),n)=(1−n/k+max{n·(n/k h_(i)), n/k−1})^(k), where k is aparameter that determines which derivatives of the algebraic expressionare continuous, n is a unit vector normal to the object at the point ofreflection, h_(i) is a unit vector bisecting an angle subtended by aunit vector pointing towards light source i from the point of reflectionand a unit vector pointing towards the observer from the point ofreflection, and n is a parameter that describes the shininess of theobject.
 20. The electronic entertainment system of claim 19, wherein2≦k≦n.